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Section 3.1 Exponential Functions and Their Graphs 203
71.
Ct P1.04 t
72. (a)
24,000
(a)
40
0 25
0
(b)
0
20
C10 35.45
(c) C10 23.951.04 10 35.45
10
(b)
t
V
t
1 2 3 4 5
17,978 13,483 10,112 7584 5688
6 7 8 9 10
V
4266 3200 2400 1800 1350
(c) According to the model, Vt → 0 as t increases.
However, V 0.
73. True. f x 1 x is not an exponential function.
74. False. e is an irrational number.
75. The graph decreases for all x and has positive y-intercept. Matches (d).
76.
y 8
1 e x (a) y
y 1 e x increases at the fastest rate.
1
y 2 x 2
y y
5 2
y4
(b) For any positive integer n, e x > x n
y 3 x 3
−9
9
large. That is, e x grows faster than
y 3
y 4 x
y 5 x
−4
for x sufficiently
x n .
(c) A quantity is growing exponentially if its growth
rate is of the form y ce rx . This is a faster rate than
any polynomial growth rate.
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77. f x 1 0.5 and gx e 0.5 1.6487
x x
78. y 3 x (c) and y 2 x (d) are exponential
functions because the exponents are variable.
(Horizontal line)
79.
81.
3
f
g
f
−3
3
−1
e
>
e
As x →, f x → gx.
e
23.14, e
22.4680.
5 3 0.008, 3 5 0.0041 82.
5 3 > 3 5
2 10 1024, 10 2 100
2 10 > 10 2
4 12 2, 1 2 4 1 16
4 12 > 1 2 4